1.1 Motivation
The observation of nature which constitutes an experiment will almost inevitably take the form of a measurement. Measurement is represented as the precision type related as whether the experiment is effective, or in the other words, how much is taken about its confidence corresponding to the experiment.
Does the measurement merely have the purpose of standing for a qualitative conclusion? Such a question causes the focus of the meaning of any experiment whenever it is significant not only for someone’s special idea but also lay themselves open to all the frailties of human judgments. That is, confidence report is needed in the formal measurement report. According to the requirement of duplicate verifications for the results of any new approach, the workers expect to convey the experimental results to someone else based on the condition of laboratory or field testing invariantly so that the level of confidence must be also included in the measurement task. Besides, confidence plays the key role to support whether to accept the other’s report so as to avoid performing a duplicate experiment. Thus the center problem for measurement task includes showing the confidence level about the results.
The best qualification of measurement is admitted as a statement of the result of human’s observations with high confidence. Because of this fundamental role of measurement it is necessary to consider in some detail what a measurement practically is. That is, how much confidence does we believe in the observations?
Why does the measurement task pay attention to the confidence factor associated with the practical experiment? According to the scientific revolution, we think that the
“uncertainty principle” brings the reason for any measurement event, especially in micro-electronics ones. For the reason to overcome the uncertainty representation, the Physics Laboratory of National Institute of Standards and Technology (NIST) conducts the standards and measurement method for electronic, optical and radiation technology for US. and takes the general Type A or Type B expression as the report for measurement task.
NIST keeps the policy based on the approach to expressing uncertainty in measurement recommended by the CIPM1and the evaluation given in the Guide to the Expression of Uncertainty in Measurement (GUM), which was prepared by individuals nominated by the BIPM, IEC, ISO, or OIML. GUM is the most authorized reference on the general application to express measurement uncertainty till 1995.
After that time, the Joint Committee for Guides in Metrology (JCGM) collects the above document and releases the new methods and standards for measurement. Thus this study will keep the work to follow the document published by JCGM as the reference. Although JCGM spent a long time for the general expression of uncertainty measurement, there are some occasions not included for practical applications and we focus on those which measure the combined signals on sparse data condition.
1.2 Stating the Function for Coverage Interval
Signal processing is a basic technique to process the sensoring signal and further sends the processed signal to the next stage or outputs it. In addition to choose a proper singal processing technique, we also need some other tools to check the properties of the input signal, such as coverage interval (CI), normal range, and reference interval, in order to determine whether the input signal is quantified to take the utility. CI is the predicative interval including a measured random quantity based on a pre-specifyied proportion of population. It is frequently applied to the cases with normal population assumption where they take the minimal CI to replace all other possible values of CI. The principal function of CI is to state the confidence and uncertainty about the measured quantities. It defines the prediction interval of values where 95% of the population fall into as suggested by JCGM [33]. For instances, we may reject the outlier data from the measured signal if the data are away from the mean value grater than 2 times of the standard deviation. A risk representation can also be applied by the way of CI to make a reject decision on sampling data if its value is out of the CI extent.
CIPM: International Committee for Weights and Measures BIPM: International Bureau of Weights and Measures IEC: International Electrotechnical Commission ISO: International Organization for Standardization OIML: International Organization of Legal Metrology
Fig. 1: Measurement is the front stage of signal processing for quantifying data recognized
Although CI is used as the standard item for the JCGM format of measurement tasks, there still have some shortcomings not being overcomed so far. The most commomly encountered problem is that CI is usually evaluated based on the assumption that the population has a asymptotically symmetric pdf, but we know this is not always appropriate, especially at the occasion of combined signal. The other CI computation method is the non-parametric method which is constructed basing on the percentile evaluated by the expectation of order statistics [1]; that is, we may take the quantile mapping to the corresponding percentile as the desired endpoint. The main difficulty of using CI for combined signal is that we don’t know whether the symmetry property of the output signal is valid when applying the CI computng algorithm. There are still other statistical techniques, such as logarithm transform and Box-Cox transformation, suggested for enhancing the symmetry properties of the analyzed signal and the outliers examining are also necessary.
1.3 Goal and Scope
Combined signal is one of the most popular measured signals for the practical usage and is widely applied to the field test as well as to the industry production. In GUM, a combined signal is represented by an additive model in which the pdf of the output
signal is modeled as the result of a propagation of input pdfs. Some special areas concern the measurement task of combined signals and treat it as an integration of the affecting factors caused from the environment.
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Fig. 2: Un-determined properties of the output pdf resulting from combining different input pdfs.
In accordance with the report expression of JCGM 101, coverage interval (CI) with its two endpoints, mean value and standard uncertainty are the three members of its main concern. They are also the main concern of this study. Due to the fact that the output of combined signal is random, we think the best description for CI representation is to formulate its pdf . Issues addressed are briefing as follows. First, we are interested in the formulation to unify the CI representations for skew and non-skew pdfs. Conventionally, different approaches are employed for these two types of pdfs to calculate their respective CI. Besides, we are also interested in the truncated probability density function normalized to its coverage. The non-skew, asymptotically symmetric pdf draws our special attention because it is the typical output pdf shape of combined signals. Moreover, the usual evaluation of asymptotically symmetric CI involves the interval composing of an upper quantile (half coverage) and a lower quantile (half coverage) with respect to the mean value. A robust CI estimation needs accurate quantile and mean formulations. This is the rule followed in the past studies so is the current study. There are a few exceptions to the rule. One is that we can consider giving a robust CI before the mean estimation, and this may leads to a good performance for mean estimation. A study will hence be conducted to try to use the traditional coverage interval to assist in the mean estimation. The issue is that if we are giving a more accurate coverage interval, can we make some progress on improving the mean estimation? Besides, we will
introduce three new approaches of mean estimation and compare them with the classical sample mean estimator. They include a quantile-based mean estimator using the coverage interval, a nonlinear mean estimator and a robust statistical one using the minimax principle. Lastly, we will shape the proposed quantile-based mean estimator to a quasi-symmetric quantile-based one and use it in an application to find the upper bound of the maximum eigenvalue (UBE), to examine the usage of the robust JCGM expression in measurement.
Fig. 3: This study reverses the traditional direction for CI estimation respect to the asymptotically symmetry pdf and further extending CI for mean estimation